Question

Use

A=P(1+rn)nt
A=P(1+rn)nt where:
A = the amortized amount (total loan/investment amount over the life of the loan/investment)
P = the initial amount of the loan/investment
r = the annual rate of interest
n = the number of times interest is compounded each year
t = the time in years

Find how long it takes a $2,200.00 investment to earn $210.00 in interest if it is invested at 10% compounded monthly.

It will take ___ years. (Round answer to 3 decimal places.)

Answer 1

same here, the interest earned is 210, the principal is 2200, so the accumulated amount is 2410.


`\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\to &\$2410\\ P=\textit{original amount deposited}\to &\$2200\\ r=rate\to 10\%\to (10)/(100)\to &0.10\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\to &12\\ t=years \end{cases}`


`\bf 2410=2200\left(1+(0.1)/(12)\right)^(12t)\implies \cfrac{2410}{2200}=\left(1+(1)/(120) \right)^(12t) \\\\\\ \cfrac{241}{220}=\left(\cfrac{121}{120} \right)^(12t)\implies log\left( (241)/(220) \right)=log\left[ \left((121)/(120) \right)^(12t) \right]\\\\\\ log\left( (241)/(220) \right)=12t\cdot log\left((121)/(120) \right)\implies \cfrac{log\left( (241)/(220) \right)}{12 log\left((121)/(120) \right)}=t\implies 0.915 \approx t`

so, about 10 months and 27 days.